Integrality of $v$-adic Multiple Zeta Values

Abstract

In this article, we prove the integrality of $v$-adic multiple zeta values (MZVs). For any index $\mathfrak{s}\in\mathbb{N}^r$ and finite place $v\in A := \mathbb{F}\_q\[\theta]$, Chang and Mishiba introduced the notion of the $v$-adic MZVs $\zeta\_A(\mathfrak{s})\_v$, which is a function field analogue of Furusho's $p$-adic MZVs. By estimating the $v$-adic valuation of $\zeta\_A(\mathfrak{s})\_v$, we show that $\zeta\_A(\mathfrak{s})\_v$ is a $v$-adic integer for almost all $v$. This result can be viewed as a function field analogue of the integrality of $p$-adic MZVs, which was proved by Akagi–Hirose–Yasuda and Chatzistamatiou.